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An Integrated, Quantitative Introduction to theNatural Sciences, Part 1: Dynamical Models1

1Notes from the Integrated Science course for first year undergraduate studentsat Princeton University. The course is an alternative to the introductory physicsand chemistry courses, and is intended to encourage students to think quanti-tatively about a much broader set of phenomena (including, especially, the phe-nomena of life) than in the usual introductory examples. We assume that stu-dents are comfortable with calculus, and have had some exposure to the ideasand vocabulary of high school physics and chemistry. This module about dynam-ical models comes in the first half of the Fall semester. All inquires can be ad-dressed to [email protected]; current students should refer tohttps://blackboard.princeton.edu for up–to–date course materials.

To the students

Last updated September 1, 2008

As we hope becomes clear, our point of view in this course is rather di!erentfrom that expressed in conventional introductory science courses. One con-sequence of this is that we can’t simply send you to a standard textbook.These lecture notes, then, are meant something that approximates a book,or more precisely a set of books, plus an extra volume for the labs. You’llsee that even the first volume of this project is far from finished. We hopethat, as with the rest of the course, you’ll view this as a collaborative e!ortbetween students and faculty, and give us feedback on what is missing fromthe notes, or on what needs to be improved.

Relation to the lectures

These notes are not an exact record of the lectures, not least because wehope the lectures (and the lecturers) are still alive enough to be evolvingfrom year to year. We do try to cover the same topics, though, in more orless the same order. Because the match between lecture notes and lecturesis loose, we suspect that these notes are not a substitute for the notes whichyou would take during the lectures. On the other hand, knowing that someof the details are written down here means that you don’t have to worryquite so much about writing down every word or equation.

The rough match between lectures and notes, for this first part of thecourse, including some material to be covered primarily in precepts, is asfollows:

0. Introduction0.1. A physicist’s point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 12 Sep0.2. A biologist’s point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . .M 15 Sep0.3. A chemist’s point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . W 17 Sep

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iv TO THE STUDENTS

1. Newton’s laws, chemical kinetics, ...1.1 Starting with F = ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 19 Sep1.2 Boxes and arrows to di!erential equations . . . . . . . . . . . . . M 22 Sep1.3 Radioactive decay and the age of the solar system . . . . .W 24 Sep1.4 Using computers to solve di!erential equations . . . . . . . . (precept)1.5 Simple circuits and population dynamics . . . . . . . . . . . . . . (precept)1.6 The complexity of DNA sequences . . . . . . . . . . . . . . . . . . . . . F 26 Sep

2. Resonance and response2.1 The simple harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . M 29 Sep2.2 Magic with complex exponentials . . . . . . . . . . . . . . . . . . . . . . W 1 Oct2.3 Damping, phases and all that . . . . . . . . . . . . . . . . . . . . . . . . . . . F 3 Oct2.4 Linearization and stability . . . . . . . . . . . . . . . . . . . . . . M 6 & W 8 Oct2.5 Stability in a real genetic circuit . . . . . . . . . . . . . . . . . . . . . . . F 10 Oct2.6 The driven oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M 13 Oct2.7 Resonance in the cell membrane . . . . . . . . . . . . . . . . . . . . . . . . . 15 Oct

3. We are not the center of the universe3.1 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .M 20 Oct3.2 Conservation of !P and !L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .W 22 Oct3.3 Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . . . F 24 Oct3.4 Universality of gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M 8 Dec3.5 Kepler’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tu 9 & W 10 Dec3.6 Biological counterpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Th 11 Dec

Problems

We cannot possibly overemphasize the importance of derivation as opposedto memorization. Science is not a long list of facts, but rather a structurefor relating many di!erent observations to one another (more about thissoon). Correspondingly, it is not enough for you to learn to recite thingsthat we show you in the lectures; we want you to develop the skills to derivethings for yourself, to develop an understanding of how di!erent things areconnected. In this spirit, we do not want to present a seamless narrative.Rather we want you to pause regularly in the reading of these notes andwork things out for yourselves. This is the role of problems, as well as someoccasional asides. Thus, instead of collecting the problems at the ends ofchapters, as many textbooks, we embed the problems at their appropriatepoints in the text, encouraging you to stop and think, pick up a pen (or,as needed, put fingers to keyboard), and calculate. Some of the problems

v

are small, essentially asking you to be sure that you understand somethingwhich goes by a bit quickly in the text. Others are longer, even open–ended,asking you to explore rather than to find a specific answer.

Students in our course should not be afraid of the large numberof problems they find here! We will assign only a fraction of the prob-lems in the weekly problem sets, which will be announced on the blackboardweb site. The problems which we don’t assign may nonetheless prove useful.Short problems could serve as warmup exercises for the assigned problems,while longer problems might serve as fodder for review as exam times ap-proach.

We take this opportunity to remind you that we encourage collaborationamong the students in working on the problems, and more generally inlearning the material of the course.

Authors

The freshman course evolved through discussions among many faculty. Thelectures which form the basis for these notes on dynamics have been givenin previous years by William Bialek, David Botstein, John Groves, MichaelHecht, and Joshua Rabinowitz; this year WB, DB & JR are joined by RobertProd’homme. Much has been added to the presentation by Michael Desaiand Matthias Kaschube, who have led precepts. Because the problems playsuch a central role in the course, we especially thank all of those students whosu!ered through the early versions, and the succession of teaching assistantswho have improved the problems as they prepared solution sets. The textas compiled here is due to WB.

vi TO THE STUDENTS

Chapter 0

Introduction

Last updated September 1, 2008

La filosofia e scritta in questo grandissimo libro che continua-mente ci sta aperto innanzi a gli occhi (io dico l’universo), manon si puo intendere se prima non s’impara a intender la lin-gua, e conoscer i caratteri, ne’ quali e scritto. Egli e scritto inlingua matematica, e i caratteri sono triangoli, cerchi, ed altrefigure geometriche, senza i quali mezi e impossibile a intenderneumanamente parola; senza questi e un aggirarsi vanamente perun’oscuro laberinto.

G Galilei 1623

Galileo’s remarks usually are paraphrased in English as “the book ofnature is written in the language of mathematics.” Indeed, one way tophrase the goal of physics is that we want to provide a concise and compellingmathematical description of nature, reading and summarizing the grandbook to which Galileo refers. Literally the hope is that everything we seearound us can be derived from a small set of equations. It is remarkablehow much of the world has been “tamed” in this way: We really do knowthe equations that describe much of what happens around us, and in manycases we actually can derive or predict what we will see by starting withthese equations and making fairly rigorous mathematical arguments. Thisis an extraordinary achievement.

We must admit, however, that much of what fascinates us in our immedi-ate experience of the world remains untamed by mathematics. In particular,life seems much more complex than anything we find in the inanimate world,and correspondingly we suspect that it will be much more di"cult to arrive

1

2 CHAPTER 0. INTRODUCTION

at convincing mathematical theories of biological phenomena. To some ex-tent this suspicion is correct, and our current understanding of biology ismuch more descriptive and qualitative than is our understanding of the tra-ditional core areas of physics and chemistry. It is important to emphasize,however, that the conventional ways of teaching greatly exaggerate these dif-ferences. Thus one teaches physics and chemistry (especially to biologists)by presenting only the very simplest of examples, and one teaches biologyby suppressing any role played by quantitative or theoretical analyses inestablishing what we actually know. For a variety of reasons, it is time forthis to change.

There is a widespread sense that we stand on the threshold of a genuinelynew chapter in our understanding of nature, and that our current qualitativedescriptions of biological phenomena will be replaced by compelling mathe-matical theories that are tested and refined through sophisticated quantita-tive experiments and computational analyses. These extraordinary scientificopportunities require a proportionately radical rethinking of our approachto undergraduate education, and this course is a first e!ort in this direction.Our goal is to present a more unified view of the sciences as a coherentattempt to discover and codify the orderliness of nature.

It should be clear that, in our view, the current divergence between theteaching of physics and the teaching of biology is a historical artifact whichshould be remedied. At the same time, the development of separate cul-tures in the di!erent disciplines is something we should appreciate, respect,and try to understand. But we must always distinguish the current stateof the culture from our ultimate goals. This distinction is not universallyaccepted. A distinguished 20th century biologist (who shall remain name-less here) also quotes Galileo, but he asserts that Galileo was limited tothe science of his time, and since he didn’t know much biology he of coursecouldn’t realize that biology would be di!erent, and presumably not math-ematical. In contrast, we see no reason to doubt Galileo’s original assertionthat understanding ultimately will mean mathematical understanding.

When we first o!ered this course in the Fall of 2004, we thought that thegoal of unification was so critical that we should stamp out any referenceto the historical traditions of di!erent disciplines. This now seems a littlenaive. Even if we agree that, as a community of scientists, we would liketo arrive someday at a seamless understanding of both the animate andinanimate worlds, and that we want this understanding to be faithful toGalileo’s vision, today the di!erent disciplines are in very di!erent places.As a result, what a biologist sees as the development of a more quantitativebiology can be quite di!erent from what a physicist sees as the physics

0.1. A PHYSICIST’S POINT OF VIEW 3

of life, and chemists would have yet a di!erent view. These di!erencesare interesting and important, and the tension among the advocates of thedi!erent points of view (perhaps even among your professors in this course!)is a creative tension. Thus, while we want to prepare you for a more unifiedview of the natural world, we want you to understand the context out whichthe next generation of scientific developments will come. To do this, westart the course with three lectures that present three di!erent viewpoints:from physics, from chemistry and from biology.1

0.1 A physicist’s point of view

The search for a compelling mathematical description of nature has led tothe invention of new mathematical structures; indeed, starting with calcu-lus, much of advanced mathematics has its origins in e!orts to understandnature. One way to organize our exploration is to identify the major classesof mathematical ideas that we use in thinking about the world, and this isthe organization that we will follow in this course. But it makes sense tostart by outlining the traditional content of a freshman physics course.

Classical mechanics. This is how we predict the trajectory of a ballwhen we throw it, and how we understand and predict the motions of theplanets around the sun; at it’s core are Newton’s laws. If we broaden ourview a bit to include the mechanics of fluids and solids, then this is also thebranch of physics that governs the motion of cilia and flagella and much ofbiological motion on the scale of microns in and around cells; at the otherend of the size scale, this is the physics that explains why planes can fly,why bridges support the weight of cars and trucks, and why the weatherchanges, sometimes unpredictably. In its original form, classical mechanicsis the great example of describing the dynamics of the world in terms ofdi!erential equations, and it is for this purpose that calculus was invented.

Thermodynamics and statistical physics. This part of our subject pro-vides laws of astonishing generality, constraining what can and cannot hap-pen even if we don’t understand all the mechanistic details—e.g., the im-

1In addition, there is the practical fact that you continue your education not as ‘inte-grated scientists’ (whatever that might mean), but as physicists, chemists, biologists, .... We want to prepare students with a coherent introduction to the natural sciences thatwill serve them well no matter which discipline they choose for their major. Thus, thisis emphatically not just a physics or chemistry course for biology majors, but rather analternative path into all of the majors. A few brave students have even used the courseas a springboard to non–science majors, and one could argue that our integrated view isespecially useful for people whose science education will end after their freshman year.


Figure 1: Not the mathemati-cal tools we had in mind whenwe designed the course.

possibility of perpetual motion machines. This also is the branch of physicsthat builds a bridge from the image of atoms and molecules whizzing aboutat random to the orderly phenomena that we see on a macroscopic scale.This is the physics of transistors and liquid crystals, of dramatic phenomenasuch as superconductivity and superfluidity, and even of the phase transi-tions that may have driven the extremely rapid expansion of our universein its initial moments. Here the underlying mathematical structure is prob-abilistic: What we see in the world are samples drawn at random out of aprobability distribution (as when we flip a coin or roll dice) and the theoryspecifies the form of the distribution.

Electricity and magnetism. It is a remarkable fact that objects seemto interact with each other over long distances, for example two magnets.In the 1800s these interactions were codified by thinking of each object asgenerating a field that pervades the surrounding space, and then the secondobject responds to the field. Electric and magnetic fields are among theearliest examples of this sort of description, and eventually it was realizedthat the equations for these fields predict a dynamics of the fields themselves,independent their sources. This dynamics corresponds to propagating waves,


much like the classical waves on the water’s surface, but the velocity ofpropagation turned out to be the speed of light, and we now understand thatlight is an electromagnetic wave. Thus electricity and magnetism providesboth a great example of how to describe the world in terms of fields anda dramatic example of unification among seemingly disparate phenomena,from the lodestone to the laser.

These three great divisions of our subject thus illustrate three di!erentstyles of mathematical description: Dynamical models in terms of di!er-ential equations, probabilistic models, and models where the fundamentalvariables are fields. One of the key ideas that we hope to communicate inthis course is that these styles of mathematical description are applicablefar beyond their origins. In particular, important parts of chemistry andphysics share this underlying mathematical structure, and the same struc-tures are applicable to the more complex phenomena of the living world.Thus instead of organizing our thinking around the historical divisions ofphysics, chemistry and biology, we will present our understanding of theworld as organized by these mathematical ideas.

Dynamical models. Newton’s laws predict the trajectories of objects asthe solutions to di!erential equations. Strikingly similar di!erential equa-tions arise in describing the kinetics of chemical reactions, the growth ofbacterial populations, and the dynamics of currents and voltages in electri-cal circuits. Rather than just teaching these separate subjects, we want togive you an appreciation for the generality and power of these ideas as aframework for understanding dynamical phenomena in nature. We won’tstop with the traditional examples, all carefully chosen for their simplic-ity, but will emphasize that the same approach describes complex networksof biochemical reactions in cells, the rich dynamics of electrical activity inneurons and networks, and so on. In order to meet these goals we will intro-duce as early as possible the art of approximation and the use of numericalmethods as ways of getting both exact answers and better intuition.

Probabilistic models. Freshman physics and freshman chemistry eachtackle the conceptually di"cult problems of thermodynamics and the statis-tical description of heat. The mathematical models here are probabilistic—when we make measurements on the world we are drawing samples out ofa probability distribution, and the theory specifies the distribution. ButMendelian genetics also is a probabilistic model, and related ideas permeatemodern approaches to the analysis of large data sets. Starting with genetics,we will introduce the ideas of probability and proceed through a rigorousview of statistical mechanics as it applies to the gas laws, chemical equi-librium, etc.. Entropy will be followed from its origins in thermodynamics


to its statistical interpretation, through its role in information theory andcoding, highlighting the startling mathematical unity of these diverse fields.We will introduce the ideas of coarse–graining and approximation, explain-ing how the same formalism applies to ideal gases and to complex biologicalmolecules.

Fields. While electromagnetism provides a compelling example of fielddynamics, the coupled spatial and temporal variations in the concentrationof di!using molecules also generate simple field equations. These equationsbecome richer as we include the possibility of chemical reactions. Surpris-ingly similar equations describe the migrations of bacterial populations inresponse to variations in the supply of nutrients, and related ideas describeproblems in ecology, epidemiology and even economics (the e–sciences).Even simple versions of the reaction–di!usion problem have important im-plications for how we think about the emergence of spatial patterns (and,ultimately, body structure) in embryonic development.

Quantum mechanics. The themes of dynamics, fields and probabilitycome together in the quantum world. Rather than a descriptive “modernphysics” course, we will present a concise account of the Schrodinger equa-tion, wave functions and energy levels, aiming at a rigorous derivation ofatomic orbitals that provides a foundation for discussing the periodic table,the geometry of chemical bonding and chemical reactivity. At the sametime we will present some of the compelling paradoxes of quantum mechan-ics, where we have a unique opportunity to discuss the sometimes startlingrelationship between mathematical models and experiment.

Reductionism vs. emergence

There is yet another way of organizing our exploration of scientific ideas, andthis is the idea of reductionism. Roughly speaking, there is a view of physicsand chemistry which says that we start with mechanics of the objects thatwe see around us, and then we start to take these apart to find out abouttheir constituents. Once we discover that matter is made from molecules,and that molecules are built from atoms, we take apart the atoms to findelectrons and the nucleus. Chemistry gives way to atomic physics, and thento nuclear physics as we try understand how the nucleus is built from itsconstituent parts, the protons and neutrons. Along the way we learn thatprotons and neutrons are not unique; if they smash into each other withenough energy we see many more particles, each with its own unique setof properties, and indeed many of these exotic particles occur naturally incosmic rays that rain down on us constantly. Nuclear physics gives way to


particle physics, and eventually we understand that the “subatomic zoo” canbe ordered by imagining that there are yet more basic constituents calledquarks; similarly the electron has several cousins in the particle zoo thatare called leptons. The interactions among these particles are described byfields—just as in electricity and magnetism, and indeed there are profoundmathematical connections among the equations for the electromagnetic fieldand the equations for the fields that mediate forces on this subatomic scale.Again the hope is for unification, and to a remarkable extent this has beenachieved.

The march from atoms to quarks in one century is one of the greatchapters in human intellectual history.2 At the same time, describing physicsonly as the constant drive to peel away layers of description, searching for the‘fundamental’ components of the universe, is a bit simplistic. In some circlesthis reductionist drive is emulated—trying to make some area of science“more like physics” often seems to mean making it more reductionist. Atthe same time, for many people “reductionist” is sort of a dirty word: Surelywe must recognize that systems are more than the sum of their parts, andthat essential functions are lost when we tear them to bits in the process ofunderstanding them.

In fact, the portrait of physics as a reductionist enterprise misses abouthalf of what physicists have been doing since !1950. The other half ofphysics is about how macroscopic or collective properties emerge from theinteractions among more elementary constituents. A familiar (if, in the end,somewhat complicated) example is provided by water. We know that purewater is made from only one kind of molecule, and the essence of the reduc-tionist claim is that the properties of water are determined by its molecularcomposition. But the most obvious properties of liquid water—-it feels wet,and it flows—clearly are not properties of single water molecules. It’s noteven clear that a small cluster with tens of molecules would have theseproperties; when we look at small numbers of molecules, for example, ourattention is immediately drawn to the fact that there is a lot of empty spacein between the molecules, while on a human scale this space is imperceptibleand the water seems to be a continuous substance.

The discrepancy between single molecules and macroscopic properties iseven greater if we think about solid water (ice). The statement that an icecube is solid is a statement about rigidity: if we push on one side of the icecube, the whole cube moves together, and in particular the opposite face

2As new Princetonians, you can take pride in knowing that some of the important stepsin this grand march took place right here.


of the cube also moves. This property can’t even be defined for a singlemolecule, and it certainly isn’t something that happens if we just have a fewmolecules. Indeed, it’s not obvious why it works at all. Imagine lifting aone kilogram cube of ice. You grip the edges, apply a force, and the entireblock is raised by say one meter. In the process you have put about 10Joules of energy into work against the force of gravity. This is more thanenough energy to rip the first layers of water molecules o! of the faces ofthe block, but that’s not what happens. Instead of flying apart in responseto the force, or flowing around your fingers like liquid water, all of the watermolecules in the block of ice (even the ones that your not touching!) movetogether. This rigidity or solidity of ice clearly has something to do withthe water molecules, but clearly involves the whole block of material beingsomething more than just the ”sum of the parts.”

There is a whole language for talking about the block of ice or the flow ofliquid water (or the flow of air—wind—in the atmosphere) that is di!erentfrom how we talk about molecules. Thus we can talk about a tornado asif it were an object moving across the globe, or about the hardness of theice surface, and neither of these things correspond in a simple way to thethings we might measure for individual molecules. Similar things happen inother materials, sometimes quite dramatically. Thus, while a single electronmoving through a solid might rattle around, bumping into various atomsand eventually losing its way, when we cool a hunk of metal down to verylow temperatures, all the electrons can flow together in an electrical currentthat lasts essentially forever, a phenomenon called superconductivity. Ifeach molecule in a liquid crystal display responded individually to appliedelectric fields, you could never get the bright and brilliant colors that yousee on your laptop computer; again there is some collective behavior in largegroups of molecules, and the whole is more than the sum of the parts.

The understanding of how these sorts of macroscopic, collective behav-iors emerge from the dynamics of electrons, atoms and molecules did notcome in one bold stroke, as in the high school version of science historywhere Einstein writes down the theory of relativity and suddenly everythingchanges. Instead there were many independent strands of thought, whichstarted to come together in the 1960s.3 Gradually it became clear that, at

3One of the landmarks was a lecture given in 1967 by our own Phil Anderson, publishedsome years later: PW Anderson, More is di!erent. Science 177, 393–396 (1972). The piecewas written partly in opposition to the notion that the reductionist search is somehow‘more fundamental’ than the search for synthesis, and Phil’s unique combative styles comesthrough clearly in his writing. Sociology aside, this paper gives a beautiful statement ofthe idea hinted at above, that even our language for describing things evolves as we move


least in a few cases, it was possible not just to understand that collectivee!ects happen, not just to describe these more macroscopic dynamics, buteven to understand how they can be derived, at least in outline, from some-thing more microscopic. In this process, something remarkable happened:we understood that when macroscopic collective e!ects arise, once we focuson these e!ect we lose track of many of the microscopic details. The flowof a fluid provides a good example, since the equations which describe thisflow depend on the density and viscosity of the fluid, and essentially nothingelse. All of the complicated properties of the molecules—the geometry oftheir chemical bonds, their van der Waals interactions and hydrogen bondswith each other, ... —none of this matters except to set the values of thosetwo parameters.4 The 1970s brought even more dramatic examples of such‘universality,’ including building precise mathematical connections betweenseemingly completely di!erent phenomena, such as the liquid crystals andsuperconductors mentioned above, but this is going too far for an introduc-tion.

We’ll come back explicitly to these ideas in the second half of the Fallsemester, and more generally our discussion of statistical physics will beall about how to build up from the microscopic description of atoms andmolecules to the phenomena that we observe on a human scale. Perhaps wecan even give a hint of how the ideas of universality have given physiciststhe courage to write down theories for much more complex phenomena, upto the phenomena of perception and memory in the human brain. For now,however, enough philosophy.

The simplest models

All of the models mentioned above involve (at least) some calculus. Butthere is a much simpler class of model, maybe one that is so simple we don’teven think of it is a “model of nature” in the grand sense that we use the

from one level (e.g., molecules) to the next (fluids and solids). There can be independentdiscoveries of the relevant mathematical laws at each level, and it is by no means obvioushow to move from one level to the next. Indeed, in those cases where we learn how toderive laws at one level from the dynamics ‘underneath,’ it is a great triumph.

4Actually one can do more. These parameters have units, and nobody tells you whatsystem of units to use. By adjusting your system of units, you can almost make the param-eters all be equal to one, so that there is truly nothing left of the molecular details. Thisisn’t quite right, because there is still a dimensionless ratio of parameters that combinesthe properties of the fluid with the typical spatial scale and speed of the fluid flow, butthis one number (Reynolds’ number) tells the whole story. Flows with the same Reynolds’number look quantitatively the same no matter what molecules make up the fluid.


word today: models in which one variable is just a linear function of theother. There are many familiar examples:5

• The voltage drop V across a resistor is proportional to the electricalcurrent I that flows through the resistor, V = IR. This is Ohm’s law,and R is the resistance.

• The force F that we feel when we stretch a spring is proportional tothe distance x that we stretch it, F = ""x. This is Hooke’s law, and" is called the sti!ness of the spring.6

• The charge Q on a capacitor (more precisely, the charge di!erencebetween the plates of the capacitor) is proportional to the voltagedi!erence V , Q = CV , where C is the capacitance.

• When an object moves through a fluid at velocity v, it experiences adrag force F = "#v, where # is called the drag or damping constant.

• The force of gravity is proportional to the mass of an object, F = mg(this one is special!).

There are even more examples, such as the fact that the rate at which yourco!ee cools is proportional to the temperature di!erence between the co!eeand the surrounding air. While simple, there really is a lot going on in these“laws.”

First of all, the notion that these are “laws” needs some revisiting. Ob-viously if you take a rubber band and pull, then pull some more, eventuallythe rubber band will snap (don’t blame me for the bruise if you feel com-pelled to verify this). Certainly when the band breaks Hooke’s law stopsbeing valid, but we suspect that the force stops being proportional to thedistance we have stretched long before the band breaks. These considera-tions describe what happens when we stretch the rubber band, but of coursewe can also compress it, and then we know that it can go slack, so that noforce is required to bring the ends together. See Fig 3. So in what sense isHooke’s law a law?

Statements such as Hooke’s law and Ohm’s law are very good approxi-mations to the properties of many real materials, but they are not “laws”

5Hopefully these are reminders of things you’ve learned in your high school course. Ifyou miss one, don’t worry, most will come back later in more detail.

6The symbol ! is the Greek letter “kappa,” and (below) " is “gamma,” both lower case.It’s incredibly useful to know the Greek alphabet; it’s not hard, and if you find yourselfin Athens you’ll be able to read the street signs. See Fig 2.


Figure 2: The Greek alphabet:upper case, lower case, andthe name of the letter. Fromhttp://gogreece.about.com/.We’ll usually write the lowercase sigma as #.

of universal applicability such as Newton’s F = ma. But why do theseapproximations work? Let’s think about some function, perhaps force as afunction of length for the spring, or voltage as a function of current in awire (resistor), etc.. Let’s call this function f(x). If you can see the wholefunction, it might be quite funny looking, as in Fig 4. On the other hand,if we only want to know the value of the function close to the place wherex = x0, we can make approximations (Fig 5).

We could start, for example, by ignoring the variations all together andsaying that since x is close to x0, we’ll just pretend that the function isconstant and equal to f(x0). A bit silly, perhaps, but maybe not so bad.The next best thing is to notice that you can fit a straight line to the functionin the neighborhood of x0, which is the same as writing

f(x) # f(x0) + a(x" x0), (1)

where a is some constant that measures the slope of the line.The next approximation would be to fit a curve in the neighborhood of

x0, starting with a parabola,

f(x) # f(x0) + a(x" x0) + b(x" x0)2. (2)

Clearly we could keep going, using higher and higher order polynomials totry and describe what is going on in the graph. Notice that by writing ourapproximations in this way we guarantee that they give exactly the rightanswer at the point x = x0.


Figure 3: Force F that opposes lengthening of a rubber band, versus the length x of theband. Straight blue line indicates Hooke’s law, F = !x. At some critical length the bandwill snap; leading up to this it takes extra force, although it’s not obvious exactly whathappens, but after the snap it takes no force to move the ends apart since they’re notconnected (!). At the other side, once the rubber band shortens to the point of goingslack, the force again goes to zero.

Some of you will recognize that what we are doing is using the Taylor se-ries expansion of the function f(x) in the neighborhood of x = x0. We recallfrom calculus that for any function with reasonable smoothness properties,for some range of x in the neighborhood of x0 we can write

f(x) = f(x0) +

!df(x)dx

""""x=x0

#· (x" x0)

+12

!d2f(x)

dx2

""""x=x0

#· (x" x0)2


0 1 2 3 4 5 6 7 8 9 10

!2

0

2

4

6

8

x

f(x) Figure 4: A function f(x), cho-

sen to have some interestingbumps and wiggles.

+13!

!d3f(x)

dx3

""""x=x0

#· (x" x0)3 + · · · , (3)

where · · · are more terms of the same general form; we can write the sameequation as

f(x) = f(x0) +!$

n=1

1n!

!dnf(x)

dxn

""""x=x0

#· (x" x0)n. (4)

Now is a good time to be sure that you remember how to read and un-derstand the summation symbol, as well as the vertical bar that means“evaluated at.”

Figure 5 makes clear that the Taylor series works in a practical sense:We can start with a pretty wild function, and if we focus our attention on asmall neighborhood then the first few terms of the Taylor series are enoughto get pretty close to the actual values of the function in this neighborhood.Notice that this practical view is di!erent from what you may have learnedin your calculus class, where the emphasis is on proving that the Taylorseries converges, i. e. that if we keep enough terms in the series we willeventually get as close as we want. The idea here is that just the first coupleof terms are enough as long as we don’t let |x" x0| get to be too big.

We can think of “laws” like Hooke’s law or Ohm’s law as the first termsin a Taylor series. Thus, as in Fig 3, the real relation between force andlength (or voltage and current) might be quite complicated, but as long aswe don’t pull or push too much, the linear approximation works. But wehaven’t said what “too much” means. Look back at Fig 4. Although thefunction has lots of bumps and wiggles, they don’t come very close together.Thus, as we sweep out a range of #x $ ["1,+1], things look pretty smooth.In fact Fig 5 focuses on a region of this size, and inside this region a loworder approximation indeed works very well. So we understand that the


4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6!1

!0.5

0

0.5

1

1.5

2

2.5

3

x

functionlinear approxquadratic approx

Figure 5: A closer view ofthe function f(x) from Fig 4in the neighborhood of x =5, together with linear andquadratic approximations.

first terms of a Taylor series are enough if we look at a range of x that issmaller than some natural scale of variations in the function we are tryingto approximate.

A key point in physical systems is that the “natural scale” we are lookingfor has units! When we say that Hooke’s law is valid if we don’t stretch thespring too much, how far is too much corresponds to some real physicaldistance. What is this distance? Similarly, when we pass current through awire, we say that Ohm’s law is valid if we don’t try to use too much currentor apply too large a voltage ... but what is the natural scale of currentthat corresponds to “too much”? Being able to answer these questions is acritical step in thinking quantitatively about the natural world, and we willreturn to these problems several times during the course.

These first problems should be thought of as warmup exercises, to get you back in therhythm of problem solving.

Problem 1: In order to answer questions about the “natural scale” for di!erentphenomena, you will need to think about orders of magnitude. It might be hard toexplain why something comes out to be exactly 347 (in some units), but you should beable to understand why it is ! 300 and not ! 30 or ! 3000. Indeed, sometimes it’s moresatisfying to have a short argument for the approximate answer than a long argument forthe exact answer.

(a.) What is the typical distance between molecules in liquid water? You should startwith the density of water, $ = 1 gram/cm3.

(b.) Many bacteria are roughly spherical, with a diameter of d ! 1 µm. If you divideup the weight of the bacterium, you find that it is 50% water, 30% protein, and 20%other molecules (e.g., RNA, DNA, lipid). A typical protein has a molecular weight of30,000 atomic mass units (or Daltons).7 Roughly how many protein molecules make upa bacterium? A typical bacterium has genes that code for about 5,000 di!erent proteins.On average, how many copies of each protein molecule is present in the cell?

7Recall that one atomic mass unit is a mass of one gram per mole.


(c.) In [b] you computed an average number of copies for all proteins, but di!erentproteins are present at very di!erent abundances inside the cell. Indeed, there are impor-tant proteins (such as the transcription factors that help to turn genes on and o!) thatfunction at concentrations of ! 1 " 10 nM. How many molecules of these proteins arepresent in the cell?

(d.) To encourage this kind of thinking, Enrico Fermi famously asked “How manypiano tuners are there in America?” during a PhD exam in Physics. Similar questionsinclude: How many students enter high school in the United States each year? How manycollege students each year need to become teachers in order to educate all these people?How many houses does the tooth fairy visit each night?8 Answer these questions, andformulate one of your own.

Problem 2: If we have a block of material with area A and length L, then thesti!ness for stretching or compressing along its length will be ! = Y A/L, where Y iscalled the Young’s modulus.

(a.) Explain why the sti!ness should be proportional to area and inversely propor-tional to length.

(b.) Show that Y has units of an energy density (or energy per unit volume—joules/m3

or erg/cm3). Note that this makes sense because the energy that we store in the blockwhen we stretch it by an amount "L, Estored = (1/2)!("L)2, works out to be proportionalto the volume (V = AL) of the block:

Estored =12!("L)2 =

12Y · (AL) ·

„"LL

«2

. (5)

(c.) Diamond is one of the sti!est materials known, and it has Y ! 1012 N/m2 (orJ/m3). The density of diamond is $ = 3.52 g/cm3. Convert Y into an energy per carbonatom in the diamond crystal. How does this compare with the energy of the chemicalbonds in the diamond crystal? Note that you’ll need to look up this number . . . be carefulabout units! Does your answer make sense?

Problem 3: Going back to the E coli in Problem 1, we want to understand theimplications of the fact that one bacterium can make a complete copy of itself (dividinginto two bacteria) in % ! 20min.

(a.) Proteins are synthesized on ribosomes, which can add ! 20 amino acids persecond to a growing protein chain. If the typical protein has 300 amino acids, how manyprotein molecules can one ribosome make within the doubling time %?

(b.) In order to double within % , the bacterium presumably has to make an extracopy of all of its protein molecules. How many ribosomes does it need in order to do this?Make use of your results from Problem 1 on the numbers of protein molecules per cell.

(c.) The ribosome is quite large as molecules go, with a diameter of ! 25 nm. If youcould cut open the bacterium and see all the ribosomes, how far apart would they be?Is there much empty space between the ribosomes, or is the cell’s interior more denselypacked?

It would be wrong to leave this discussion of very simple models withouttalking about one of the very simplest—so simple, in fact, that from our

8Admittedly, this is a bit more hypothetical than Fermi’s problem.


modern perspective we can miss that it is a model at all. You all know that

2H2 + O2 % 2H2O. (6)

We talk about this equation in terms of each water molecule being made outof two hydrogen atoms and one oxygen atom. But not so long ago, we didn’tknow about atoms. What we did know was that if you mix hydrogen andoxygen together, you get water. When people looked more carefully, theyfound that this could be made quantitative: A certain amount of hydrogenand oxygen, in certain proportions, are needed to make a certain amount ofwater. But “amounts” are measured in some units which, from our modernpoint of view, are rather arbitrary—liters (or worse, gallons) of the gases,grams of the liquids or solids, ... . So you might learn that some numberof liters of hydrogen gas plus some number of liters of oxygen gas producessome number of grams of liquid water.

44.8 liters H2 + 22.4 liters O2 % 36 grams H2O. (7)

Now you do other experiments, mixing hydrogen and oxygen with othermaterials. Each time there is some rule about how the di!erent amountcombine. It really was an amazing discovery that if you choose your unitscorrectly you can turn all these funny numbers [as in Eq (7)] into the integersof Eq (6). Thus, if you say that one unit of gas at room temperature andpressure is 22.4 liters, then all of the reactions involving gases simplify, andso on.

What’s going on here? Now we know, of course, that we should measurethe number of molecules or the number of moles of each substance, andthen the rule for combining macroscopic quantities just reflect the rules forcombining individual atoms. But we can think about trying to write downthe rules for the macroscopic quantities alone, and then these are simplefunctional relations—in fact they are linear relations, not unlike Hooke’slaw. The wonderful thing is that the coe"cients in these linear relationsdon’t take on arbitrary values (as with the sti!ness of a spring) but if wechoose our units correctly these coe"cients are just pure numbers. It tooka long time to go from this discovery to the modern view of atoms andmolecules, but when you find that there is a way of looking at the world inwhich the numbers you need to know are integers, you know you’re on tosomething!

What physicists do

I’d like to add a little more here, to bring the text back to whatphysicists do today ...

0.2. A CHEMIST’S POINT OF VIEW 17

0.2 A chemist’s point of view

0.3 A biologist’s point of view

Josh & David, would you like to put something here in print?

an integrated, quantitative introduction to the …wbialek/intsci_web/dynamics0.pdfan integrated,...

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